Hereditarily non uniformly perfect non-autonomous Julia sets

Mark Comerford; Rich Stankewitz; Hiroki Sumi

doi:10.3934/dcds.2020002

Previous Article
Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line
DCDS Home
This Issue
Next Article
Stationary states of the cubic conformal flow on $ \mathbb{S}^3 $

January 2020, 40(1): 33-46. doi: 10.3934/dcds.2020002

Hereditarily non uniformly perfect non-autonomous Julia sets

Mark Comerford ^1, , Rich Stankewitz ^2, and Hiroki Sumi ^3,

1.	Department of Mathematics, University of Rhode Island, 5 Lippitt Road, Room 102F, Kingston, RI 02881, USA
2.	Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA
3.	Course of Mathematical Science, Department of Human Coexistence, Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto 606-8501, Japan

Received October 2018 Revised June 2019 Published October 2019

Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [19] who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of $ 1 $ for these Julia sets can be attained. The proof of the latter considers the Julia set as the limit set of a non-autonomous conformal iterated function system and we calculate the Hausdorff dimension using a version of Bowen's formula given in the paper by Rempe-Gillen and Urbánski [15].

Keywords: Hereditarily non uniformly perfect sets, non-autonomous iteration.

Mathematics Subject Classification: Primary: 30D05; Secondary: 28A80.

Citation: Mark Comerford, Rich Stankewitz, Hiroki Sumi. Hereditarily non uniformly perfect non-autonomous Julia sets. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 33-46. doi: 10.3934/dcds.2020002

References:

[1]	L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics.
[2]	Francisco Balibrea, On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci., 1 (2016), 391-404. doi: 10.21042/AMNS.2016.2.00034.
[3]	E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2008.
[4]	L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.
[5]	M. Comerford, A survey of results in random iteration, Proceedings Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), 435–476.
[6]	M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynamical Systems, 26 (2006), 353-377. doi: 10.1017/S0143385705000441.
[7]	A. Eremenko, Julia Sets are Uniformly Perfect, Preprint, Purdue University, 1992.
[8]	K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications.
[9]	J. E. Fornæss and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708. doi: 10.1017/S0143385700006428.
[10]	A. Hinkkanen, Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc., 113 (1993), 543-559. doi: 10.1017/S0305004100076192.
[11]	S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., (4) (1996), 205-233.
[12]	R. Mañé and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116 (1992), 251-257. doi: 10.1090/S0002-9939-1992-1106180-2.
[13]	C. T. McMullen, Complex Dynamics and Renormalization, Volume 135 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994.
[14]	C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740. doi: 10.1007/s00039-010-0078-3.
[15]	L. Rempe-Gillen and M. Urbański, Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017. doi: 10.1090/tran/6490.
[16]	O. Sester, Hyperbolicité des polynȏmes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428.
[17]	R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575. doi: 10.1090/S0002-9939-00-05313-2.
[18]	R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589. doi: 10.3934/dcds.2012.32.2583.
[19]	R. Stankewitz, H. Sumi and T. Sugawa, Hereditarily non uniformly perfect sets, Discrete Contin. Dyn. Syst S, 12 (2019), 2391-2402. doi: 10.3934/dcdss.2019150.
[20]	H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019. doi: 10.1088/0951-7715/13/4/302.
[21]	H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603. doi: 10.1017/S0143385701001286.
[22]	H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922. doi: 10.1017/S0143385705000532.
[23]	H. Sumi, Dynamics of postcritically bounded polynomial semigroups Ⅲ: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902. doi: 10.1017/S0143385709000923.
[24]	W. Zhiying, Moran sets and Moran classes, Chinese Sci. Bull., 46 (2001), 1849-1856. doi: 10.1007/BF02901155.

show all references

References:

[1]	L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics.
[2]	Francisco Balibrea, On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci., 1 (2016), 391-404. doi: 10.21042/AMNS.2016.2.00034.
[3]	E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2008.
[4]	L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.
[5]	M. Comerford, A survey of results in random iteration, Proceedings Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), 435–476.
[6]	M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynamical Systems, 26 (2006), 353-377. doi: 10.1017/S0143385705000441.
[7]	A. Eremenko, Julia Sets are Uniformly Perfect, Preprint, Purdue University, 1992.
[8]	K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications.
[9]	J. E. Fornæss and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708. doi: 10.1017/S0143385700006428.
[10]	A. Hinkkanen, Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc., 113 (1993), 543-559. doi: 10.1017/S0305004100076192.
[11]	S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., (4) (1996), 205-233.
[12]	R. Mañé and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116 (1992), 251-257. doi: 10.1090/S0002-9939-1992-1106180-2.
[13]	C. T. McMullen, Complex Dynamics and Renormalization, Volume 135 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994.
[14]	C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740. doi: 10.1007/s00039-010-0078-3.
[15]	L. Rempe-Gillen and M. Urbański, Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017. doi: 10.1090/tran/6490.
[16]	O. Sester, Hyperbolicité des polynȏmes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428.
[17]	R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575. doi: 10.1090/S0002-9939-00-05313-2.
[18]	R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589. doi: 10.3934/dcds.2012.32.2583.
[19]	R. Stankewitz, H. Sumi and T. Sugawa, Hereditarily non uniformly perfect sets, Discrete Contin. Dyn. Syst S, 12 (2019), 2391-2402. doi: 10.3934/dcdss.2019150.
[20]	H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019. doi: 10.1088/0951-7715/13/4/302.
[21]	H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603. doi: 10.1017/S0143385701001286.
[22]	H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922. doi: 10.1017/S0143385705000532.
[23]	H. Sumi, Dynamics of postcritically bounded polynomial semigroups Ⅲ: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902. doi: 10.1017/S0143385709000923.
[24]	W. Zhiying, Moran sets and Moran classes, Chinese Sci. Bull., 46 (2001), 1849-1856. doi: 10.1007/BF02901155.

Figure 1. How the survival sets $ {\mathcal S}_k $ are nested. The pictures show preimages of $ \overline {\mathrm D}(0,2) $ at stages $ M_k $ (in red) and $ M_{k-1} $ (in blue) with $ m_k = 3 $. The dashed blue circle is $ {\mathrm C}(0,2) $ while the unit circle is shown in black. Observe how $ Q_{M_{k-1},M_k}^{-1}(\overline {\mathrm D}(0,2)) \subset \overline {\mathrm D}(0, 2) \setminus \overline {\mathrm D}(0, 1) $ as in Remark 1(c) is shown in red at Stage $ M_{k-1} $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Schematic for the proof of Theorem 1.6 in the case where $ \limsup |c_k| = +\infty $. Note how the round annulus $ {\mathrm A}(\sqrt{-c_{k}}, 1, \sqrt{|c_{k}|}) $ at stage $ M_{k-1} + m_k $ (in this case $ M_1+m_2 $) is pulled back conformally first by the preimage branches of $ Q_{M_{k-1},M_{k-1}+m_k} $ to form half the members of the collection $ \mathcal C $ at Stage $ M_1 $. Then the preimage branches of $ Q_{M_{k-1}} $ pull back the annuli in $ \mathcal C $ (one of which is visible in the zoomed box) to conformal annuli which separate the components of $ \mathcal{S}_k $ at stage $ 0 $

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Rich Stankewitz, Toshiyuki Sugawa, Hiroki Sumi. Hereditarily non uniformly perfect sets. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : 2391-2402. doi: 10.3934/dcdss.2019150
[2]	Mark Comerford, Todd Woodard. Orbit portraits in non-autonomous iteration. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : 2253-2277. doi: 10.3934/dcdss.2019144
[3]	Mark Comerford. Non-autonomous Julia sets with measurable invariant sequences of line fields. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 629-642. doi: 10.3934/dcds.2013.33.629
[4]	Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703
[5]	Michael Khanevsky. Non-autonomous curves on surfaces. Journal of Modern Dynamics, 2021, 17: 305-317. doi: 10.3934/jmd.2021010
[6]	Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[7]	JinHyon Kim, HyonHui Ju, WiJong An. Inheritance of $ {\mathscr F}- $chaos and $ {\mathscr F}- $sensitivities under an iteration for non-autonomous discrete systems. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022053
[8]	Thorsten Hüls, Yongkui Zou. On computing heteroclinic trajectories of non-autonomous maps. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 79-99. doi: 10.3934/dcdsb.2012.17.79
[9]	Thorsten Hüls. A model function for non-autonomous bifurcations of maps. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 351-363. doi: 10.3934/dcdsb.2007.7.351
[10]	Maciej J. Capiński, Piotr Zgliczyński. Covering relations and non-autonomous perturbations of ODEs. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 281-293. doi: 10.3934/dcds.2006.14.281
[11]	Pablo G. Barrientos, Abbas Fakhari. Ergodicity of non-autonomous discrete systems with non-uniform expansion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1361-1382. doi: 10.3934/dcdsb.2019231
[12]	Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020
[13]	Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[14]	Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067
[15]	Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082
[16]	Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701
[17]	Cung The Anh, Tang Quoc Bao. Dynamics of non-autonomous nonclassical diffusion equations on $R^n$. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1231-1252. doi: 10.3934/cpaa.2012.11.1231
[18]	Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17
[19]	Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003
[20]	Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743

2020 Impact Factor: 1.392

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.392

Tools

Figures and Tables

2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

Hereditarily non uniformly perfect non-autonomous Julia sets

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Hereditarily non uniformly perfect non-autonomous Julia sets

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors